For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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2
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2answers
71 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
1
vote
2answers
42 views

induction on matrices with powers + addition and limit

$A= \begin{bmatrix} 1-q && p \\ q && 1-q \end{bmatrix}, 0<p<1, 0<q<1,$ Using mathematical induction show that $A^n$ = $\frac{1}{p+q}\begin{bmatrix} q && p \\ q ...
1
vote
2answers
54 views

Standard matrix A of T?

Help please. What would be the standard matrix of A? I know how to do number 2 and 3 but I'm just having trouble with A. I asked this earlier but I lost my account and I'm not sure if I posted ...
1
vote
1answer
21 views

A question about matrix spectrum property

Suppose $x\in\mathbb{R}^n$, $A\in\mathbb{R}^{n\times n}$. Does anyone know the answer to the following problems. (1) $\min\limits_{x\neq0} f(x)=\frac{x^\mathrm{T}A^\mathrm{T}Ax}{x^\mathrm{T}Ax}$, ...
1
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1answer
59 views

Another matrix of a given operator $A \otimes A$

Let $V$ be a $4$-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $(e_{1}, ...
1
vote
1answer
72 views

Is this function involving matrices convex?

Let $X\in \mathbb{R}^{n \times n}$. Then, is the function $$ \text{Tr}\left( (X^T X )^{-1} \right)$$ convex in $X$? ($\text{Tr}$ denotes the trace operator)
1
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1answer
26 views

Estimate $\frac{(\frac{1}{2}\lambda_{\text{min}}(A+A^T))^2}{\lambda_{\text{max}}A^TA}<1$

If $A$ is positive definite, (maybe not symmetric), how to prove that $$\frac{(\frac{1}{2}\lambda_{\text{min}}(A+A^T))^2}{\lambda_{\text{max}}A^TA}<1,$$ I know that ...
1
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3answers
63 views

How to get An eigenvalue and eigenvectors of a matrix that contain both zero column and zero row?

Could anyone help in how to get the eigenvalue and eigenvectors of a matrix that contain both zero column and zero row like : \begin{pmatrix} -1 & 1 & 0\\ 1 & -1 & 0\\ 0 ...
0
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3answers
47 views

For two p.d. matrices $A$ and $B$, prove that $\lambda_1(AB)\leqslant \lambda_1(A) \cdot\lambda_1(B)$

If $A$ and $B$ are two nxn positive definite matrices, then show that $$\lambda_1(AB) \leqslant \lambda_1(A) \cdot \lambda_1(B),$$ where $\lambda_1(\cdot)$ denotes the largest eigenvalue.
4
votes
1answer
66 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
1
vote
1answer
115 views

Does Fermat's Little Theorem apply to matrices?

I'm working on a problem involving applying FLT to matrices, so any information about how to do this or prove this is true would be helpful. I've been doing some research and experimenting a little, ...
1
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2answers
79 views

Find the rank of the given matrix

Let $x_1$,$x_2$,$x_3$,$x_4$,$y_1$,$y_2$,$y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a 4 x 4 matrix A by A = $$\begin{pmatrix} x_1^2 + y_1^2 & x_1x_2 + y_1y_2 ...
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2answers
83 views

The eigenvalue of $A^TA$

If $\lambda$ is the eigenvalue of matrix $A$,what is the eigenvalue of $A^TA$?I have no clue about it. Can anyone help with that?
0
votes
1answer
285 views

Find unknown matrix in equation with 3 multiplications.

A matrix $D$ is calculated as $A*B*C$. I need to find the matrix $B$ given matrices $A$, $C$ and $D$. After some trial and error it seems that the following equation is needed to reproduce matrix ...
2
votes
1answer
97 views

Determining the structure of the abelian group, integral matrix

I am revising for my upcoming university exams and I have a past exam question that I am finding particularly challenging... a) Consider the integral matrix $$R=\begin{bmatrix} 2 & 2 & ...
0
votes
1answer
265 views

Using QR decomposition to solve a system of equations with a singular matrix

If $A\in\mathbb{R}^{n\times n}$ is singular and $x,b\in\mathbb{R}^{n}$ are such that $Ax=b$, am I right in thinking that the upper triangular matrix $R$ of $A$'s $QR$ decomposition must have at least ...
0
votes
1answer
33 views

SOR method converges for $\left( \begin{array}{ccc}2& -1\\-2 & 2\end{array} \right)$

Prove that the SOR method converges in $\mathbb{R}^n$ for the matrix $\left( \begin{array}{ccc}2& -1\\-2 & 2\end{array} \right)$ iff $\omega\in(0,2)$.
0
votes
1answer
22 views

Retrieving a Matrix from a Matrix multiplication

I have made a matrix multiplication in Matlab (K, P and S are all 2x2 matrices): K = P * transpose(H)*S Now Im given K, P, and H. I need to know S. Given that ...
2
votes
1answer
46 views

Prove the matrix $\left( \begin{array}{ccc}A_{11}&A_{12}\\A_{21}&B_{22}+A_{21}A_{11}^{-1}A_{12}\end{array}\right)$ spd

Let $$A=\left( \begin{array}{ccc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right)\in R^{n\times n}$$ be a symmetric positive definite matrix with blocks $A_{ij}\in\mathbb{R}^{n_i\times ...
2
votes
1answer
19 views

least square approximation: how this matrix calculation equation is deducted?

I am reading a book "kernel methods for pattern analysis". For the least square approximation, it is to minimise the sum of the square of the discrepancies: $$e=y-Xw$$ Therefore it is to minimize $$ ...
0
votes
1answer
62 views

Prove that $F$ is dense in $C(X\times Y,\mathbb{R})$?

Let $X$ and $Y$ be compact metric spaces. Let $$ F= \Bigl\{\sum_{i=1}^n A_i f_i(x) g_i(y): f_i\in C(X,\mathbb{R}),g_i\in C(Y,\mathbb{R}), 1\le i\le n \Bigr\}. $$ Prove that $F$ is dense in $C(X\times ...
2
votes
1answer
264 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
2
votes
1answer
61 views

One step Gauss Seidel method

Apply one step of the Gauss Seidel method to $A\textbf{x} = b$ with A = $\begin{bmatrix} 4 & 2 & 1 \\ 1 & 4 & 1 \\ 1 & 2 & 4 \end{bmatrix}$, b = $\begin{bmatrix} 4\\ ...
2
votes
1answer
154 views

How does a cropping of a 2D matrix/image affect its DCT transform?

I apologize in advance: since I am not a mathematician, maybe my question is not well defined, but I hope that some of you will still understand my meaning. Given a 2D matrix, or an image of ...
0
votes
2answers
62 views

Uniqueness in Matrix Multiplication

I'm sure there is an answer to this somewhere else, but I'm simply not sure how to find it or what to call it. I looked online, but couldn't find anything. The question is as follows: Let $A$ and ...
0
votes
1answer
97 views

Constructing a complete affine 3D transformation matrix with homogeneous coordinates.

I have been able to scale, rotate, and translate a 2D point represented by a 3x1 matrix as such: $$P = \left( \array{ x \\ y \\1 } \right)$$ The affine transformation that I apply to $P$ is this ...
5
votes
1answer
53 views

Matrix Help: Combinations

Given a 10 by 10 matrix filled with 0s and 1s, how many possible outcomes are there? It sounds easy enough as a combination of $2^{100}$. The kicker to the question is there MUST be exactly five 1's ...
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0answers
35 views

Basic Question Linear Transformation and Matrix computations

Can someone show me how to do this question? http://imgur.com/cIciHnY I'm studying for a test and this was a question off a past test. I would love to show my thoughts but I do not know how to format ...
0
votes
1answer
36 views

Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates

For some experimental and practical reason, I have created a new coordinate system in the form $$x^\prime_i=T_{ij}x_j$$ where $T_{ij}$ isn't a square matrix. $x_i$ is standard Cartesian coordinates, ...
0
votes
1answer
47 views

Linear maps and matrix coefficients

I am currently working through this page in my script: Can somebody explain what this means and how it works in practice? Perhaps if I saw an example I could follow it. Thanks for your help!
0
votes
1answer
25 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
1
vote
2answers
35 views

Looking for notation of set of all entries of some matrix?

I'm busy writing my thesis, and I'm looking for some concise notation to denote the supremum of the matrix entries of, say $A \in M_n(\mathbb{R})$. How should I do this? Looking for something like ...
2
votes
2answers
87 views

When do eigenvectors converge?

Let $A_n$ be a sequence of self-adjoint $N\times N$ matrices that converge in the operator norm to $A$. The sequence of eigenvalues of $A_n$, denoted $\lambda_n$, converges to an eigenvalue of $A$, ...
5
votes
1answer
999 views

Anti-commutative matrices

If $A$ and $B$ are anti-commutative square matrices, so $AB+BA=0$, how do you a) prove that $\mathrm{tr}(A)=\mathrm{tr}(B)=0$ and b) prove that the order of the matrices is even?
4
votes
1answer
429 views

Spectral radius and positive definite of matrices

Denote $ \rho(A)$ to be the spectral radius of a matrix $A,$ that is the maximal eigenvalue of $A.$ We say that a matrix $M$ is positive definite, respectively positive semidefinite, if $x^TMx>0$ ...
5
votes
4answers
334 views

if $AB\neq 0$ for any non zero matrix $B$ then $A$ is invertible

Question is to check that : If $A$ is an $n\times n$ matrix over a field $F$ and $AB\neq 0$ for any non zero matrix $B_{n\times n}$ over $F$ then, $A$ is invertible. This does make some sense to me ...
1
vote
1answer
252 views

A is a Hermitian projection if and only if it is an orthogonal projection

I need to figure out this property of Hermitian / Orthogonal projections "A is a Hermitian projection if and only if it is an orthogonal projection" Your assistance will be highly appreciated. ...
0
votes
1answer
35 views

On the eigenvalues / properties of a specific matrix.

I'm not sure how to better phrase the title of the question, because I don't know the specific name of the matrix I am after, but I want to consider matrices of the form $$ \begin{align*} ...
0
votes
0answers
59 views

Find Determinant of linear transformation

The question is Find the determinant of linear transformation Let V be the vector space of polynomials of degree at most over R, and define T:V to V by T(p(x))=p(1+x)-p'(1-x) for all p(x) in V. I ...
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0answers
38 views

A question about eigenvalues of a special block matrix

Thanks for anyone who views or answers this question! $N\times N$ matrix $L$ can be partitioned into $p\times p$ blocks, ...
0
votes
1answer
24 views

Question regarding change of basis

Given two basis {$\textbf{e}_a$} and {$\textbf{e}_{a'}$}, we can have $$\textbf{e}_a = R^{b'}_a\textbf{e}_{b'}$$ $$\textbf{e}_{a'} = R^{b}_{a'}\textbf{e}_{b}$$ Substituting the second equation into ...
0
votes
1answer
31 views

Question about Symmetric matrix

Ok my book says this matrix $A = \left ( \array{ -2 & 1 \\ 1 & -3 } \right )$is symmetric. But, I don't understand b/c if it were a symmetric matrix, wouldn't it be ...
1
vote
3answers
136 views

Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
0
votes
1answer
28 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
1
vote
1answer
32 views

c0mpatible system $A^TAx=A^Tb$

Let $A\in\mathbb{R}^{n\times n}$ be a singular matrix. Prove that the system $$A^TAx=A^Tb$$ is compatible for any $b\in\mathbb{R}^n$. I want to prove that $A^Tb\in Ran(A^TA)$,i.e. $A^Tb\bot ...
1
vote
0answers
104 views

Shortened Generator Matrix

goodmorning, could someone tell me if the following code has been handled correctly? I have this generator matrix (which I should modify in order to have it correct): $$G=\begin {bmatrix} ...
0
votes
0answers
30 views

Proving that $\mathrm{rank}(P_1+P_2) = \mathrm{rank}(P_1)+\mathrm{rank}(P_2)$

Supposing $P_1$ and $P_2$ two projectors as: $P_1\circ P_2 = P_2\circ P_1$. What is the condition for $P_1+P_2$ to be a projection? If it was the case above then how can I prove that ...
2
votes
2answers
72 views

Showing that matrix admits an eigenvector?

Let A= a b c d be a 2 x 2 matrix, where a,b,c and d are real numbers. We say that A admits an eigenvector if there exists a unit vector u and a real ...
2
votes
1answer
69 views

Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
2
votes
1answer
38 views

How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

I have the following matrix $$ \begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix} $$ First I got the eigenvalues which are $0$, $1$, $2$. I tried to get the ...